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13
votes
1answer
727 views

Does every sheaf embed into a quasicoherent sheaf?

Question. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$?...
2
votes
1answer
164 views

Computation of cohomology of ideal sheaves

Let $j: X \to Y$ be a closed embedding. Let $I_{X/Y}$ be the ideal sheaf of this closed embedding. Then there is a exact sequence $$ I_{X/Y} \to \mathcal{O}_Y \to j_{*}\mathcal{O}_X \to 0$$ One use ...
1
vote
1answer
133 views

What is the hypercohomology of the push-forward of the intersection chain complex of an open cone to its closure?

Let $X = \left(L \times [0, 1]\right) / \left(L \times \{0\}\right)$ be the closed cone over a closed smooth $d$-dimensional manifold $L^{d}$. Let $i \colon Y \hookrightarrow X$ denote the inclusion ...
2
votes
0answers
118 views

Defineing a Sheaf of rings over a topological space

Let $X$ be a topological space and let $R$ be a commutative ring with $1$ such that for each $x\in X$ there exists a multiplicatively closed subset $S_x$ of $R$ such that for each $a\in R$ if $\frac{a}...
3
votes
0answers
76 views

Semicontinuity of cohomology of torsion-free sheaves restricted to divisors

Let $X$ be a smooth projective variety, $\mathcal{E}$ a torsion-free coherent sheaf on $X$ and $\mathfrak{d}$ a linear system of divisors in $X$. I would like to show (at least when $X$ is a surface) ...
1
vote
1answer
231 views

Reformulation of Grothendieck vanishing theorem

Let $X$ be a smooth, projective variety, ${F}$ a quasi-coherent $\mathcal{O}_X$-module on $X$ supported on a closed subscheme, say $Z \subset X$. Is it true that $H^i(X,F)=0$ for all $i>\dim Z$? ...
1
vote
1answer
129 views

Sheaf cohomology of a complement of finitely many points

Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $p \subset X$ a closed point in $X$. How do I compute $H^1(\mathcal{O}_{X\backslash p})$? Any reference/idea will be most welcome.
1
vote
0answers
65 views

Thom-type isomorphism on sheaf cohomology

Let $X$ be a smooth, projective surface and $T$ a finite set of points in $X$ i.e., of codimension $2$ in $X$. Is it true that $H^i(\mathcal{O}_X)=H^i(\mathcal{O}_{X\backslash T})$ for $i \ge 1$?
0
votes
1answer
146 views

Dual of a stable locally free subsheaf is a locally free quotient sheaf

Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$. By [1] definition 1.2: A line bundle $L$ ...
5
votes
1answer
219 views

Naive question on local cohomology

Let $X$ be a smooth, projective variety and $Z_1, Z_2$ two smooth, projective subvarieties in $X$ of the same dimension. Let $E$ be a locally free sheaf on $X$. Recall, there are natural morphims: $$...
-1
votes
0answers
200 views

Is chapter 5 of Grothendieck (1955) related to Sheaf Cohomology?

I'm curious if the fifth and last chapter of Grothendieck's 1955/1958 University of Kansas research report General theory of fibre spaces with structure sheaf (Grothendieck Circle pdf copy; link to ...
3
votes
0answers
113 views

What is the filtration in Leray's spectral sequence?

Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration $$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...
1
vote
0answers
50 views

Continuous maps vs filtrations construction of the Leray spectral sequence

The Leray spectral sequence of a continuous map $f : X \to Y$ between two topological spaces can be constructed, as far as I understand, in two ways, one very general, and the other a bit more ...
0
votes
1answer
160 views

Sheaf cohomology relative to a closed subspace

Let $i : A \hookrightarrow X$ be a closed subspace of a topological space $X$, and $j : Z := X \setminus A \hookrightarrow X$ denote its open complement. Given a sheaf $F$ of abelian groups on $X$, ...
1
vote
1answer
131 views

Commutativity between functors on sheaves of abelian groups

I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f :...
1
vote
0answers
156 views

How to calculate : $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) $?

I try to calculate the rational cohomology algebra $ \mathrm{Hdg}^{ 2 \bullet } ( \mathcal{H}\mathrm{ilb} ( \mathbb{P}^n ),\mathbb{Q} ) = \displaystyle \bigoplus_{k=0}^{+ \infty} \mathrm{Hdg}^{ 2 k } (...
3
votes
0answers
67 views

Convolution of sheaves on R

I am trying to understand a basic computation of convolution. Throughout, $R$ is the real line as a topological group and $k$ is some base field. I would like to understand the computation of the ...
2
votes
1answer
242 views

sheaf cohomology and deRham cohomology of a stack

I am reading https://arxiv.org/pdf/math/0605694.pdf to understand about (hyper)cohomology groups of stack $\mathcal{X}$ with valued in a complex of abelian sheaves $\mathcal{M}$. Let $\mathcal{F}$ be ...
1
vote
0answers
52 views

maps between two Leray spectral sequences based on maps on cochain complexes

Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...
1
vote
0answers
96 views

Maps between Leray spectral sequences

Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps between topological spaces. Suppose that there exists a commutative diagram of singular cohomology groups (say with coefficients ...
2
votes
1answer
110 views

Leray spectral sequence for continuous functions on pairs of topological spaces

Let $A \subset X$ and $B \subset Y$ be topological spaces, with $A$ and $B$ closed, and $f: X \to Y$ be a continuous functions such that $f(A) \subset B$. The Leray spectral sequence (with complex ...
5
votes
1answer
219 views

On the Leray spectral sequence and sheaf cohomology

I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two ...
3
votes
0answers
57 views

exact sequence of fundamental groups associated to “almost” smooth families of curves

Suppose I have a proper, flat family of curves $X \to S$ that has a section. Fix a basepoint $s \in S$ and let $X_s$ denote the corresponding fiber. Let $\mathbb{L}$ be a set of primes which does not ...
2
votes
0answers
123 views

An algorithm to compute coherent sheaf cohomology in projective space over a ring [closed]

EDIT: As the article was put on hold, because it was unclear what I am asking, here I put again my two questions: 1) Is the argument I used to derive the algorithm valid? The second question is a ...
3
votes
0answers
55 views

Canonical map in the direct image of $\mathscr{D}_X$

Let $f : X \to Y$ be a proper holomorphic map between holomorphic manifolds. We work with $\mathscr{D}$-modules. Consider the transfer bi-modules $\mathscr{D}_{Y\leftarrow X}.$ Can one find a ...
5
votes
1answer
221 views

Sheaf cohomology with support vanishes

I am trying to solve the exercise 2.4 chapter III in Hartshorne's "Algebraic Geometry". For this I would like to prove for a sheaf $F$ of Abelian groups on a topological space $X$ and $U$ open subset ...
1
vote
1answer
510 views

Inception of modern view of Sheaf Cohomology in Mathematical Literature

From wikipedia entry on Sheaf Cohomology I have found the intriguing passage: 'The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups ...
0
votes
0answers
64 views

English translation of Faisceaux algebriques coherents J-P.S 1955 [duplicate]

Grothendieck (1957; 3. Cohomology with coefficients in a sheaf) mentions this paper [Faisceaux algebriques coherents] of J-P.Serre as fundamental. I'd really appreciate if anyone could point me to an ...
0
votes
1answer
131 views

The cohomology of meromorphic functions

Let $A$ be a sheaf such that $$A(U) = \{ f \in \mathbb M(U): f \in \mathbb{O}(U \backslash\{p_1,\ldots, p_n\}) \ \mbox{with at worst a simple pole at}\ p_i \} $$ where $\mathbb M(U)$ means the set of ...
2
votes
0answers
121 views

Reference for the Koszul--Malgrange Theorem

The Koszul--Malgrange theorem, roughly, identifies holomorphic vectors bundles over a complex manifold, as those finitely generated projective modules admitting a flat $(0,1)$-connection. The ...
5
votes
1answer
173 views

Cohomology of sheaf of Schwartz distributions with support in a submanifold

Let $M$ be a smooth manifold. Let $Z\subset M$ be a smooth submanifold which is a closed subset. Let $F$ denote the sheaf of generalized functions (equivalently, Schwartz distributions) on $M$, namely ...
1
vote
0answers
171 views

Cohomology of a structure sheaf of a normal affine variety

I can't find the reference for the following fact: Let $X$ be an affine variety and let $Y$ be its smooth resolution. $H^0(X,\mathcal{O}_x)=H^0(Y,\mathcal{O}_Y)$ if and only if $X$ is normal.
1
vote
1answer
214 views

Cartier Divisor generated by Global Sections

Let $X$ be an integer curve of (arithmetic) genus $g=0$. (the arithmetic genus $g$ is defined by $g:= 1 -\chi_k(\mathcal{O}_X)$ where $\mathcal{O}_X$ is the structure sheaf of $X$ and $\chi_k(\mathcal{...
2
votes
2answers
108 views

Continuous map with homeomorphic fibers whose associated $H^{k}_c$ sheaf is not a local system?

Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$. Let $k$ be a ring and for every $...
3
votes
0answers
84 views

Question on de Rham complex with distributional coefficients

Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...
13
votes
1answer
441 views

Is there a complex which computes Cech cohomology?

Suppose $X$ is a (paracompact, Hausdorff) topological space and we want to define its Cech cohomology with coefficients in $\mathbb Z$. Here is the way I have seen this constructed. For each open ...
2
votes
1answer
276 views

Is any element in $H^2_{et}(X,\mathcal{O}_X^*)$ locally trivial in the Zariski topology?

Let $X$ be an algebraic variety over a field $k$ and we consider the cohomological Brauer group $H^2_{et}(X,\mathcal{O}_X^*)$. For any element $\alpha \in H^2_{et}(X,\mathcal{O}_X^*)$ and any closed ...
7
votes
1answer
306 views

What are the uses of coefficient systems for arithmetic cohomology theories?

In topology when studying a space with non-trivial fundamental group it becomes important to consider homology and cohomology with coefficients in representations of the fundamental group, i.e. local ...
2
votes
0answers
83 views

Theta divisor on compactified jacobian of nodal curve

Let $X$ be a Nodal curve. Let $\bar{J}(X)$ be compactified Jacobian (rank one torsion free sheaf of degree one) and $\Theta$ denote the theta divisor in $J$. How to compute $H^0(\bar{J}(X);\Theta^k)$, ...
6
votes
0answers
168 views

Defining the Euler class in different ways

Let $\pi: E\to M$ be a rank two real vector bundle over a manifold $M$. Bott and Tu defines the Euler class by: giving $M$ a Riemannian structure, taking a trivializing chart $U_\alpha$ of $M$, ...
8
votes
0answers
104 views

Countability assumption for good covers in Bott-Tu

In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....
8
votes
0answers
286 views

Leray-Hirsch theorem for Dolbeault cohomology

In Bott and Tu's Differential forms in algebraic topology there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this: Theorem (Leray-Hirsch): Let $E$ be a fiber bundle over $M$...
8
votes
1answer
515 views

Do the cohomology groups of the structure sheaf of a smooth resolution depend on the resolution?

Let $X$ be an affine variety. Let $Y$ be smooth and let the map $f\colon Y\rightarrow X$ be proper birational. We will call $Y$ a smooth resolution of $X$. Do the cohomology groups $H^i(Y,\mathcal{O}...
1
vote
0answers
71 views

Local cohomology with supports in a constructible set

Let $X$ be a topological space (I'm interested in the case of $X$ being a complex algebraic variety with the Zariski topology) and $Z$ a constructible subset (i.e. a finite union of locally closed ...
3
votes
0answers
65 views

Cohomology of boundary of locally symmetric space

Let $S$ be a locally symmetric space, not necessarily compact, and $\overline{S}$ be its Borel-Serre compactification. Let $\partial S$ be the boundary of $S$. Let $\widetilde{\mathbb{C}}$ be the ...
2
votes
0answers
95 views

Cohomology of adelic locally symmetric spaces

I am most probably wrong in asserting as follows. Let $G$ be a connected reductive group over $\mathbb{Q}$, and $S_{K_f} = G(\mathbb{Q}) \backslash G(\mathbb{A}/K_\infty Z(\mathbb{A}) \cdot K_f $ be ...
5
votes
0answers
183 views

Does the de Rham complex induce a functorial soft resolution of the category of cochain complexes of sheaves of vector spaces on a smooth manifold?

I apologize in advance if this is pretty straightforward; I'm a differential geometer and physicist by training so my homological algebra and homotopy theory are a bit weak. Question: Let $M$ be a ...
3
votes
0answers
135 views

$\operatorname{Ext}^2(O,\omega)$ as a higher extension on $\mathbb{P}^1 \times \mathbb{P}^1$

Let $X = \mathbb{P}^1 \times \mathbb{P}^1$ over a field $k$ and consider $Ext^2(\mathcal{O}_X,\omega_X)\cong H^2(\omega_X) = H^2(\mathcal{O}_X(-2,-2)) = k$ Let $C = \mathbb{P}^1$. By Kunneth $H^2(\...
6
votes
0answers
271 views

What is the geometric intuition for the sheaf-theoretic terms “soft”, “fine”, and “flabby”?

The sheaf-theoretic terms "soft", "flabby", and "fine" are of an obviously geometric character, and suggest opposition with "hard", "rigid", and "coarse" sheaves (I'm just inventing these terms here). ...
9
votes
0answers
336 views

In terms of sheaf cohomology, what does Bott & Tu's relative de Rham cohomology $H^\bullet(f)$ compute for $f: S \to M$ a smooth map?

Given a map $f: S \to M$ of smooth manifolds, Bott & Tu define on page 78 a complex by $\Omega^q(f)=\Omega^q(M) \oplus \Omega^{q-1}(S)$ and $d(\omega, \theta)=(d\omega, f^*\omega - d\theta)$ where ...